IMC 2018 Q19
19. The three rectangles shown below all have the same area.
What is the value of \(x + y\)?
- A 4
- B 6
- C 8
- D 10
- E 12
Intermediate Mathematical Challenge 2018, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 10
As the three rectangles have equal areas: \(xy = (x + 4)(y - 3) = (x + 8)(y - 4)\).
So \(xy = xy - 3x + 4y - 12 = xy - 4x + 8y - 32\).
Therefore \(-3x + 4y - 12 = 0\) and \(-4x + 8y - 32 = 0\).
Rearranging: \(3x - 4y = -12\) … (1) and \(4x - 8y = -32\) … (2).
2 × (1) − (2) gives: \(6x - 4x = -24 - (-32) = 8\). So \(x = 4\).
Substituting for \(x\) in (1): \(12 - 4y = -12\). So, \(4y = 24\), that is \(y = 6\).
So \(x + y = 4 + 6 = 10\).
Solution © UK Mathematics Trust, from the IMC 2018 solutions. Longer versions with investigations: Solutions and investigations (PDF).